math
On beauty bare
Wang tiles
Trees and graphs
A tree is a kind of graph, but a graph can be considerably more complex than a tree.
I have reason to believe, which for brevity’s sake I will treat elsewhere, that the most complex class of processes and structures we humans can consciously prescribe, reduces mathematically to a tree. A tree has a top, bottom, left and right. Its branches fan out from the trunk and they don’t intersect with one another. They are discrete, contiguous, identifiable objects which persist across time. Trees are Things.
Software and websites, however, reduce to arbitrarily more complex structures: they are graphs. A graph has no meaningful orientation whatsoever. No sequence, no obvious start or end—at least none that we can intuit. It is better considered not as one Thing, but as a federation of Things, like the brain or a fungus network, or perhaps a composite artifact left behind from an ongoing process, like an ant colony or human city.
Trees and semilattices
The tree of my title is not a green tree with leaves. It is the name of an abstract structure. I shall contrast it with another, more complex abstract structure called a semilattice.
Both the tree and semilattice are ways of thinking about how a large collection of many small systems goes to make up a large and complex system.
A collection of sets forms a semilattice if, and only if, when two overlapping sets belong to the collection, the set of elements common to both also belongs to the collection. That is, if [234] and [345] belong to the collection, then [34] belongs to the collection.
A collection of sets forms a tree if, and only if, for any two sets that belong to the collection either one is wholly contained in the other, or they are wholly disjoint. Every tree is trivially a simple semilattice.
We are concerned with the difference between structures in which no overlap occurs, and those structures in which overlap does occur.
The semilattice is potentially a much more complex and subtle structure than a tree. It is this lack of structural complexity, characteristic of trees, which is crippling our conceptions of the city.
A City Is Not a Tree
An Essay by Christopher Alexander- Strands of life
- Impending destruction
- The right overlap
- The difficulty of designing complexity
- Political chains of influence
Notes on the Synthesis of Form
A Book by Christopher AlexanderVisualizing Data
A Book by William S. ClevelandExploratory Data Analysis
A Book by John TukeyPlus Equals #4
An Article by Rob WeychertOne of the seeds for Plus Equals was planted a few years ago with Incomplete Open Cubes Revisited, my extension of a Sol LeWitt work. I learned a lot about isometric projection from that project, but my affection for the concept didn’t begin there. Whether I’m looking at a Chris Ware illustration or an exploded-view technical drawing of a complex machine, an isometric rendering always stirs something in me.
A brief foray into vectorial semantics
An Article by James SomersOne of the best (and easiest) ways to start making sense of a document is to highlight its “important” words, or the words that appear within that document more often than chance would predict. That’s the idea behind Amazon.com’s “Statistically Improbable Phrases”:
Amazon.com’s Statistically Improbable Phrases, or “SIPs”, are the most distinctive phrases in the text of books in the Search Inside!™ program. To identify SIPs, our computers scan the text of all books in the Search Inside! program. If they find a phrase that occurs a large number of times in a particular book relative to all Search Inside! books, that phrase is a SIP in that book.
tixy.land
A Websitesin(t * x) * cos(t * y)
Creative code golfing.
Rafael Araujo's Golden Ratio
A GalleryBlue Morpho Double Helix & Icosahedron
The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy
A Research Paper by Cyril Stanley SmithA pattern of tiles illustrated by Douat in 1722.
A translation is given of Truchet's 1704 paper showing that an infinity of patterns can be generated by the assembly of a single half—colored tile in various orientations.
Everything and More
A Book by David Foster WallaceInfoCrystal
A Research PaperThis paper introduces a novel representation, called the InfoCrystal, that can be used as a visualization tool as well as a visual query language to help users search for information. The InfoCrystal visualizes all the possible relationships among N concepts.
A Search for Structure
Apologia
A ChapterGrain Shapes and Other Metallurgical Applications of Topology
An EssayStructure, Substructure, and Superstructure
An EssayThe Interpretation of Microstructures of Metallic Artifacts
An EssayMatter versus Materials: A Historical View
An EssayIconography
It is understandable that those students who must work from reproductions of works of art are usually more interested in iconography than in the more subtle questions of technique and quality, but it is regrettable that technical ignorance should so frequently prevent art historians from considering the whole experience of the artist.
Understanding technique
Technique is an essential aspect of any work of art from a trivial trinket to the greatest painting, and some specialized study of it is essential to full appreciation.
Though museum labels and catalogs refer to materials and processes — “bronze,” “fresco,” “parcel gilt,” “tempera,” “lacquer on wood,” and so on — they usually display only superficial attention to the essential details of the artist’s technique.
Bells
Most Japanese bells when hung still have on them one or more rough lines obviously arising in horizontal mold joints. These lines are not removed in fettling the bell, and they seem to be regarded not as defects but rather as a reminder of the reality of the founder’s interaction with his materials. One is reminded of the ceramics that are most treasured in Japan which usually have some unexpected tool marks or irregularity resulting from a kiln mishap.
Nearer to the surface
If in the following I overemphasize the Orient, this is simply because in the Far East the properties of materials are a little nearer to the surface, a little more consciously a part of what the artist is trying to show. The naturalistic aspects of Oriental philosophy encourage a sensitivity to the quality of materials — or is it the inverse, that an early enjoyment of stone, wood, clay, and fiber gave rise to the philosopher’s perception of the soul in all natural things comparable to man himself? Westerners tend to override materials, usually in ignorance, but sometimes proudly as a tour de force.
The drop press
The virtue of thin sheet metal in giving the greatest glitter for a grain of gold was exploited in the earliest days of metallurgy. However, before the days of rolled sheet and drawn wire, most metal objects were made by hammering and were basically three-dimensional in form.
[In contrast] look at the simple drop press — it’s unmodulated blow striking in a single direction symbolizes much of nineteenth-century mechanized production. To make multiple stampings, stacks of very thin metal sheets were superimposed under the hammer, and the final profile with moderately high relief was gradually achieved as finished sheets were removed from the bottom and new ones added at the top.
When the drop press was used to shape large areas of thin sheet metal, the aesthetic qualities of the surface became divorced from the underlying substance, and decoration became independent of the body needed to support it. In any object there is a natural relationship between the surface and the bulk, that is, between its one-, two-, and three-dimensional aspects. The fakery involved in applying gold or silver playing on a solid copper object is quite different from the deception of an ornately stamped piece of thin sheet brass. Compare a magnificent ormolu furniture fitting or even a gilded plaster picture frame with a cheap lamp base embossed in thin sheet brass. In the former, the surface is simply and honestly applied for its optical effect alone; in the latter the fakery is fundamental for it is dimensionally misleading.
Big things and little things
It is hardly possible that human beings could have decided logically that they needed to develop language in order to communicate with each other before they had experienced pleasurable interactive communal activities like singing and dancing. Aesthetic curiosity has been central to both genetic and cultural evolution.
All big things grow from little things, but new little things will be destroyed by their environment unless they are cherished for reasons more like love than purpose.
What's the difference?
I well remember an occasion in 1962 when in a remote Iranian village I asked a blacksmith famous for his superior penknives to tell me the difference between iron and steel. “What’s the difference?” he replied. “What’s the difference between an oak tree and a willow — they have different natures and one must adapt to them.” He did not accept the suggestion that some material absorbed from the fire’s charcoal might have something to do with it, and he would not have understood a word of any lecture I could have given him on diffusion, crystal structure, and phase transformations; yet he could make a good knife and I could not.
The scale of human experience
It is the scale of human experience, from which thought and imagination take off, and to which they must return.
The edifice from which they came
A list of types of bricks used in the Hagia Sophia may help one to build an interesting brick wall, but it poorly suggests the great edifice from which they came.
The interplay of pattern and texture
Some of the more enjoyable surfaces (for example, the grain of a fine mahogany table top or a Japanese sword) have an interplay between pattern and texture which, though two-dimensional, suggests the unseen internal three-dimensional array.
Resonances
The resonances arising in workmanship are often very subtle. The fact that the material itself guides the tool differently in different processes of working introduces changes in the overall relationship of curvatures. The smooth curves of surfaces approaching the edge of a jade axe that come about from innumerable abrasive particles moving against a slightly yielding and mechanically unconstrained backing would seem incongruous if other surfaces or outlines were present that had come from cleavage or from the geometric motions of a machine. These could be produced easily enough, but the eye would not establish larger resonances among them.
Fine arts and decorative arts
The fine arts are conscious and essentially individual in tradition.
The quantitative and economic aspects of the decorative arts, on the other hand, make them intrinsically repetitive. Because of this, their aesthetic qualities have a very intimate relationship to the technology of materials, and their design is thereby basically affected.
In addition to the qualitative need for repetitive detail in design, the decorative arts have a quantitative requirement, namely the imperative of covering large areas or making large numbers of individual objects.
Replication
Consider also the development of mass-production methods involving the casting of molten metals. Though the finest castings were made individually by the lost-wax process, the majority of casting from the earliest days have been designed expressly to facilitate molding. As with punches and dies, most foundry processes have the characteristic that the careful work of the master designer is involved only once, whereafter replication takes over.
Reproduction
The success of a mechanic’s, or a machine’s, reproduction of a thing depends on his, or its, sensitivity to whatever qualities are important, just as the skill of the designer lies in the proper appreciation of surface qualities in terms of structure and shape variation that come from the intended means of production.
The inner nature of material
The work of an artist in getting the details that he wants is greatly facilitated if he selects a material whose inner nature makes it want to take the desired shape.
The idea grows as they work
As they work, the experience of the material under the artist's fingers subtly interacts with the idea in their mind to give the finished work some quality that was rarely fully anticipated. A few artists seem to have such a feeling for their materials that the prevision needs little modification; most say that the idea grows as they work experimentally.
True artistry
The best of all examples of a satisfactory art form based upon the inner nature of a metal is provided by Japanese swords.
Our perception of beauty seems to involve the interaction of several patterns having origin and significance at many different levels of space, time, matter, and spirit. In the Japanese sword blade there is heterogeneity in both the macrostructure and the microstructure. The manner of forging, the heat treatment, and the final polishing operation are all uniquely Japanese techniques, and all make necessary contributions to the final quality of the blades. The shape along would be simplistic form; the forged texture of the steel without heat treatment would at best faintly echo the beauty of grained wood; the outlines of the quench-hardened zone at the edge would be sharp and uninteresting if it depended only on the control of cooling rate during quenching; and the polish would be uniform glitter if the metal were homogeneous. With true artistry all these are made to interact.
Simple variations of the parts
Symmetry, indeed, has been grossly overemphasized in both art and science: its main value is in giving meaning to its absence, dissymmetry, without which there could be no hierarchy.
The eye is repulsed by complexity if no order is detected, but it can be delighted by repetition, translation, rotation, reflection, magnification, and other simple variations of the parts.